Finite Group Databasebeta

← C2×C3×#1516 #10636175

Subgroups — C2×C3×#1516 #10636175

Order 768.

1498 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#525C211yesmaximal, characteristic
2C22×C3×C22⋊C8C211yesmaximal, characteristic, complemented
3C2×C3×#381C211yesmaximal, characteristic, complemented
4C3×#1516C218yesmaximal, direct product factor
5C2×C3×#328C212yesmaximal, complemented
6C2×C3×#327C212yesmaximal, complemented
7C2×#1516C311yesmaximal, characteristic, direct product factor, Sylow
8C22×C3×#41C411yescharacteristic, complemented
9C24×C3×C4C2211yescharacteristic
10C23×C3×C8—21no—
11C3×#525—22no—
12C3×#525—22no—
13C3×#525C414yescomplemented
14C3×#525C2214yes—
15C2×C3×#123—21no—
16C2×C3×C22⋊Q8C411yescharacteristic, complemented
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×#121C411yescharacteristic, complemented
19C2×C3×#121C2211yescharacteristic
20C2×C3×#120—21no—
21C2×C3×C4×D4—21no—
22C2×C3×C4×D4—21no—
23C2×C3×#117C411yescharacteristic, complemented
24C22×C3×C22⋊C4—21no—
25C2×C3×C22⋊C8—22no—
26C2×C3×C22⋊C8—21no—
27C2×C3×C22⋊C8C2212yes—
28C2×C3×C22⋊C8C2214yescomplemented
29C3×#381—22no—
30C3×#381C2214yescomplemented
31C2×C3×#95—21no—
32C2×C3×#95C2212yes—
33C3×#328C2218yescomplemented
34C3×#327C2218yescomplemented
35C2×#525C2×C311yescharacteristic
36C22×C22⋊C8C2×C311yescharacteristic, complemented
37C2×#381C2×C311yescharacteristic, complemented
38#1516C2×C318yesdirect product factor
39C2×#328C2×C312yescomplemented
40C2×#327C2×C312yescomplemented
41C25×C3D411yescharacteristic
42C2×C3×#41—22no—
43C2×C3×#41—44no—
44C2×C3×#41C2×C414yescomplemented
45C22×C3×Q8C2×C411yescharacteristic
46C22×C3×D4—21no—
47C22×C3×D4—21no—
48C22×C3×D4—21no—
49C22×C3×D4C2×C411yescharacteristic
50C23×C3×C4—21no—
51C23×C3×C4—22no—
52C23×C3×C4—44no—
53C23×C3×C4C2×C411yescharacteristic
54C23×C3×C4C2×C411yescharacteristic
55C23×C3×C4C2×C412yes—
56C23×C3×C4D411yescharacteristic
57C23×C3×C4D411yescharacteristic
58C23×C3×C4D412yes—
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D412yes—
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2312yes—
63C22×C3×C8—22no—
64C22×C3×C8—24no—
65C22×C3×C8—44no—
66C3×#123—24no—
67C3×#123—24no—
68C3×C22⋊Q8—22no—
69C3×C22⋊Q8—21no—
70C3×C22⋊Q8—21no—
71C3×C22⋊Q8C2×C412yes—
72C3×C22⋊Q8C2×C414yescomplemented
73C3×C22⋊Q8C2312yes—
74C3×#121—22no—
75C3×#121—21no—
76C3×#121—21no—
77C3×#121C2×C412yes—
78C3×#121C2×C414yescomplemented
79C3×#121C2312yes—
80C3×#120—24no—
81C3×#120—24no—
82C3×C4×D4—24no—
83C3×C4×D4—24no—
84C3×C4×D4—24no—
85C3×C4×D4—24no—
86C3×#117—22no—
87C3×#117C2×C414yescomplemented
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2×C411yescharacteristic
91C2×C3×C4⋊C4C2×C411yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—24no—
98C2×C3×C22⋊C4—44no—
99C2×C3×C42—21no—
100C3×C22⋊C8—22no—
101C3×C22⋊C8—24no—
102C3×C22⋊C8C2314yes—
103C3×#95—24no—
104C3×#95—22no—
105C3×#95C2314yes—
106C22×#41C3×C411yescharacteristic, complemented
107C24×C4C22×C311yescharacteristic
108C23×C8—21no—
109#525—22no—
110#525—22no—
111#525C3×C414yescomplemented
112#525C22×C314yes—
113C2×#123—21no—
114C2×C22⋊Q8C3×C411yescharacteristic, complemented
115C2×C22⋊Q8C22×C311yescharacteristic
116C2×#121C3×C411yescharacteristic, complemented
117C2×#121C22×C311yescharacteristic
118C2×#120—21no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×#117C3×C411yescharacteristic, complemented
122C22×C22⋊C4—21no—
123C2×C22⋊C8—22no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8C22×C312yes—
126C2×C22⋊C8C22×C314yescomplemented
127#381—22no—
128#381C22×C314yescomplemented
129C2×#95—21no—
130C2×#95C22×C312yes—
131#328C22×C318yescomplemented
132#327C22×C318yescomplemented
133C24×C3—24no—
134C24×C3—24no—
135C24×C3—21no—
136C24×C3—42no—
137C24×C3C22⋊C411yescharacteristic
138C24×C3C22⋊C412yes—
139C24×C3C22⋊C411yescharacteristic
140C24×C3C2×D412yes—
141C24×C3C2×D411yescharacteristic
142C3×#41—48no—
143C3×#41—48no—
144C2×C3×Q8—21no—
145C2×C3×Q8—42no—
146C2×C3×Q8C22×C412yes—
147C2×C3×D4—22no—
148C2×C3×D4—24no—
149C2×C3×D4—24no—
150C2×C3×D4—22no—
151C2×C3×D4—21no—
152C2×C3×D4—42no—
153C2×C3×D4—44no—
154C2×C3×D4—44no—
155C2×C3×D4—44no—
156C2×C3×D4C22×C412yes—
157C22×C3×C4—24no—
158C22×C3×C4—24no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—24no—
163C22×C3×C4—24no—
164C22×C3×C4—42no—
165C22×C3×C4—44no—
166C22×C3×C4—42no—
167C22×C3×C4—44no—
168C22×C3×C4—42no—
169C22×C3×C4—44no—
170C22×C3×C4—42no—
171C22×C3×C4—44no—
172C22×C3×C4—44no—
173C22×C3×C4—44no—
174C22×C3×C4—44no—
175C22×C3×C4—44no—
176C22×C3×C4C22⋊C412yes—
177C22×C3×C4C22⋊C412yes—
178C22×C3×C4C22⋊C411yescharacteristic
179C22×C3×C4C22⋊C411yescharacteristic
180C22×C3×C4C22⋊C412yes—
181C22×C3×C4C22⋊C412yes—
182C22×C3×C4C22⋊C412yes—
183C22×C3×C4C22×C412yes—
184C22×C3×C4C22×C411yescharacteristic
185C22×C3×C4C22×C411yescharacteristic
186C22×C3×C4C22×C412yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2411yescharacteristic
194C2×C3×C8—24no—
195C2×C3×C8—44no—
196C2×C3×C8—48no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4C22×C412yes—
203C3×C4⋊C4C22×C412yes—
204C3×C22⋊C4—24no—
205C3×C22⋊C4—22no—
206C3×C22⋊C4—24no—
207C3×C22⋊C4—22no—
208C3×C22⋊C4—22no—
209C3×C22⋊C4—24no—
210C3×C22⋊C4—22no—
211C3×C22⋊C4—48no—
212C3×C42—24no—
213C25C3×D411yescharacteristic
214C2×#41—22no—
215C2×#41—44no—
216C2×#41C2×C3×C414yescomplemented
217C22×Q8C2×C3×C411yescharacteristic
218C22×D4—21no—
219C22×D4—21no—
220C22×D4—21no—
221C22×D4C2×C3×C411yescharacteristic
222C23×C4—21no—
223C23×C4—22no—
224C23×C4—44no—
225C23×C4C2×C3×C411yescharacteristic
226C23×C4C2×C3×C411yescharacteristic
227C23×C4C2×C3×C412yes—
228C23×C4C3×D411yescharacteristic
229C23×C4C3×D411yescharacteristic
230C23×C4C3×D412yes—
231C23×C4C3×D411yescharacteristic
232C23×C4C3×D412yes—
233C23×C4C23×C311yescharacteristic
234C23×C4C23×C312yes—
235C22×C8—22no—
236C22×C8—24no—
237C22×C8—44no—
238#123—24no—
239#123—24no—
240C22⋊Q8—22no—
241C22⋊Q8—21no—
242C22⋊Q8—21no—
243C22⋊Q8C2×C3×C412yes—
244C22⋊Q8C2×C3×C414yescomplemented
245C22⋊Q8C23×C312yes—
246#121—22no—
247#121—21no—
248#121—21no—
249#121C2×C3×C412yes—
250#121C2×C3×C414yescomplemented
251#121C23×C312yes—
252#120—24no—
253#120—24no—
254C4×D4—24no—
255C4×D4—24no—
256C4×D4—24no—
257C4×D4—24no—
258#117—22no—
259#117C2×C3×C414yescomplemented
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4C2×C3×C411yescharacteristic
263C2×C4⋊C4C2×C3×C411yescharacteristic
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—44no—
271C2×C42—21no—
272C22⋊C8—22no—
273C22⋊C8—24no—
274C22⋊C8C23×C314yes—
275#95—24no—
276#95—22no—
277#95C23×C314yes—
278C23×C3—22no—
279C23×C3—22no—
280C23×C3—22no—
281C23×C3—22no—
282C23×C3—24no—
283C23×C3—24no—
284C23×C3—24no—
285C23×C3—22no—
286C23×C3—24no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—416no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3#9914yes—
297C23×C3C2×C22⋊C412yes—
298C23×C3C2×C22⋊C412yes—
299C23×C3C2×C22⋊C411yescharacteristic
300C23×C3C2×C22⋊C411yescharacteristic
301C23×C3#13514yes—
302C23×C3C22×D411yescharacteristic
303C3×Q8—42no—
304C3×Q8—42no—
305C3×D4—42no—
306C3×D4—42no—
307C3×D4—48no—
308C3×D4—44no—
309C3×D4—48no—
310C3×D4—44no—
311C2×C3×C4—24no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—22no—
316C2×C3×C4—24no—
317C2×C3×C4—22no—
318C2×C3×C4—24no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—41no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—41no—
329C2×C3×C4—42no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—48no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—44no—
337C2×C3×C4—44no—
338C2×C3×C4—44no—
339C2×C3×C4—88no—
340C2×C3×C4C2×C22⋊C412yes—
341C2×C3×C4C2×C22⋊C411yescharacteristic
342C2×C3×C4C2×C22⋊C412yes—
343C2×C3×C4C2×C22⋊C411yescharacteristic
344C2×C3×C4C23×C411yescharacteristic
345C2×C3×C4C22×D411yescharacteristic
346C3×C8—44no—
347C24—24no—
348C24—24no—
349C24—21no—
350C24—42no—
351C24C3×C22⋊C411yescharacteristic
352C24C3×C22⋊C412yes—
353C24C3×C22⋊C411yescharacteristic
354C24C2×C3×D412yes—
355C24C2×C3×D411yescharacteristic
356#41—48no—
357#41—48no—
358C2×Q8—21no—
359C2×Q8—42no—
360C2×Q8C22×C3×C412yes—
361C2×D4—22no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—42no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4C22×C3×C412yes—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—42no—
381C22×C4—44no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4C3×C22⋊C411yescharacteristic
391C22×C4C3×C22⋊C411yescharacteristic
392C22×C4C3×C22⋊C412yes—
393C22×C4C3×C22⋊C412yes—
394C22×C4C3×C22⋊C412yes—
395C22×C4C3×C22⋊C412yes—
396C22×C4C3×C22⋊C412yes—
397C22×C4C22×C3×C411yescharacteristic
398C22×C4C22×C3×C412yes—
399C22×C4C22×C3×C411yescharacteristic
400C22×C4C22×C3×C412yes—
401C22×C4C2×C3×D411yescharacteristic
402C22×C4C2×C3×D412yes—
403C22×C4C2×C3×D412yes—
404C22×C4C2×C3×D411yescharacteristic
405C22×C4C2×C3×D412yes—
406C22×C4C2×C3×D411yescharacteristic
407C22×C4C24×C311yescharacteristic
408C2×C8—24no—
409C2×C8—44no—
410C2×C8—48no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—22no—
416C4⋊C4C22×C3×C412yes—
417C4⋊C4C22×C3×C412yes—
418C22⋊C4—24no—
419C22⋊C4—22no—
420C22⋊C4—24no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—24no—
424C22⋊C4—22no—
425C22⋊C4—48no—
426C42—24no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—22no—
430C22×C3—21no—
431C22×C3—22no—
432C22×C3—21no—
433C22×C3—28no—
434C22×C3—28no—
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—48no—
439C22×C3—48no—
440C22×C3—44no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—84no—
445C22×C3C2×#9914yes—
446C22×C3C2×#9912yes—
447C22×C3#41612yes—
448C22×C3#41612yes—
449C22×C3#42112yes—
450C22×C3C22×C22⋊C411yescharacteristic
451C22×C3C2×#13512yes—
452C3×C4—21no—
453C3×C4—22no—
454C3×C4—22no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—42no—
459C3×C4—42no—
460C3×C4—42no—
461C3×C4—84no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—22no—
470C23—24no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—42no—
475C23—42no—
476C23—416no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23C3×#9914yes—
481C23C2×C3×C22⋊C412yes—
482C23C2×C3×C22⋊C412yes—
483C23C2×C3×C22⋊C411yescharacteristic
484C23C2×C3×C22⋊C411yescharacteristic
485C23C3×#13514yes—
486C23C22×C3×D411yescharacteristic
487Q8—42no—
488Q8—42no—
489D4—42no—
490D4—42no—
491D4—48no—
492D4—44no—
493D4—48no—
494D4—44no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—48no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—88no—
524C2×C4C2×C3×C22⋊C412yes—
525C2×C4C2×C3×C22⋊C411yescharacteristic
526C2×C4C2×C3×C22⋊C412yes—
527C2×C4C2×C3×C22⋊C411yescharacteristic
528C2×C4C23×C3×C411yescharacteristic
529C2×C4C22×C3×D411yescharacteristic
530C8—44no—
531C2×C3—22no—
532C2×C3—24no—
533C2×C3—22no—
534C2×C3—42no—
535C2×C3—82no—
536C2×C3#151614yesdirect product factor
537C2×C3C22×#9911yescharacteristic
538C2×C3C2×#41611yescharacteristic
539C2×C3C2×#42111yescharacteristic
540C22—21no—
541C22—21no—
542C22—22no—
543C22—21no—
544C22—22no—
545C22—21no—
546C22—28no—
547C22—28no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—48no—
552C22—48no—
553C22—44no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—84no—
558C22C2×C3×#9914yes—
559C22C2×C3×#9912yes—
560C22C3×#41612yes—
561C22C3×#41612yes—
562C22C3×#42112yes—
563C22C22×C3×C22⋊C411yescharacteristic
564C22C2×C3×#13512yes—
565C4—21no—
566C4—22no—
567C4—21no—
568C4—22no—
569C4—21no—
570C4—21no—
571C4—42no—
572C4—42no—
573C4—42no—
574C4—84no—
575C3C2×#151611yescharacteristic, direct product factor, Sylow
576C2—22no—
577C2—24no—
578C2—22no—
579C2—42no—
580C2—82no—
581C2C3×#151614yesdirect product factor
582C2C22×C3×#9911yescharacteristic
583C2C2×C3×#41611yescharacteristic
584C2C2×C3×#42111yescharacteristic
subgroupG/H
center, socleC23×C3C2×C22⋊C4
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#525C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C22×C3×C22⋊C8C2semidirect
3. C2×C3×#381C2semidirect
4. C3×#1516C2direct (N × H)
5. C2×C3×#328C2semidirect
6. C2×C3×#327C2semidirect
7. C2×#1516C3direct (N × H)
8. C22×C3×#41C4semidirect
13. C3×#525C4semidirect
16. C2×C3×C22⋊Q8C4semidirect
18. C2×C3×#121C4semidirect
23. C2×C3×#117C4semidirect
28. C2×C3×C22⋊C8C22semidirect
30. C3×#381C22semidirect
33. C3×#328C22semidirect
34. C3×#327C22semidirect
36. C22×C22⋊C8C2×C3semidirect
37. C2×#381C2×C3semidirect
38. #1516C2×C3direct (N × H)
39. C2×#328C2×C3semidirect
40. C2×#327C2×C3semidirect
44. C2×C3×#41C2×C4semidirect
72. C3×C22⋊Q8C2×C4semidirect
78. C3×#121C2×C4semidirect
87. C3×#117C2×C4semidirect
106. C22×#41C3×C4semidirect
111. #525C3×C4semidirect
114. C2×C22⋊Q8C3×C4semidirect
116. C2×#121C3×C4semidirect
121. C2×#117C3×C4semidirect
126. C2×C22⋊C8C22×C3semidirect
128. #381C22×C3semidirect
131. #328C22×C3semidirect
132. #327C22×C3semidirect
216. C2×#41C2×C3×C4semidirect
244. C22⋊Q8C2×C3×C4semidirect
250. #121C2×C3×C4semidirect
259. #117C2×C3×C4semidirect
536. C2×C3#1516direct (N × H)
575. C3C2×#1516direct (N × H)
581. C2C3×#1516direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×#5252
C22×C3×C22⋊C82
C2×C3×#3812
C3×#15162
C2×C3×#3282
C2×C3×#3272
C2×#15163